Graynomad:
The above sounds reasonable to get the power turned on but something has to latch it as the average doorbell press would not be long enough for the Arduino to boot and do it's job.
Unless the Arduino is just sleeping waiting for the doorbell interrupt on an appropriate pin, and not turned off. I think my UNO and the other newer Arduinos spend some amount of time at power on looking to see if there is a new program to be installed via USB/FTDI, and it might not give you the fast response that is desired.